A cantilever beam is made by connecting two rods together. The left half,(0,l/2) , of the beam has properties, E1 ,I1 and A1 , and the right half,(l/2,l) , has properties, E2 ,I2 and A. The beam is subjected to a distributed load q(x), and a horizontal load at the free end, P. The length of the beam is denoted as l. Assume that the material is linear elastic and the beam follows the linear strain-displacement relation. The beam is fixed at x=0 Thus, it is with given boundary conditions, u(0)=w(0)=w' (0)=0. The displacement field of the beam is assumed as

u(x)= uo(x)- zw'(x)
v=0
w(x)= wo(x)

Derive:
a. the total potential energy, n(u(x), w(x)), and
b. the equations of equilibrium and the associated boundary conditions at x=0, l/2 and l in terms of u(x) and w(x).